Abstract

We prove existence and uniqueness of entropy solutions for the Neumann problem for the quasilinear elliptic equation $u - \mathrm{div} \, \mathbf{a} = v$, where $v\!\in \! L^1$, $\mathbf{a} = \nabla _\xi f$, and $f$ is a convex function of $\xi $ with linear growth as $\Vert \xi \Vert \rightarrow \infty $, satisfying other additional assumptions. In particular, this class includes the case where $f = \varphi \psi $, $\varphi > 0$, $\psi $ being a convex function with linear growth as $\Vert \xi \Vert \rightarrow \infty $. In the second part of this work, using Crandall-Ligget’s iteration scheme, this result will permit us to prove existence and uniqueness of entropy solutions for the corresponding parabolic problem with initial data in $L^1$